Trommarinn brosir til söngkonunnar á meðan hún lagar fiðluna sína.

Because Icelandic usually puts the definite article on the end of the noun.

• trommari = drummer
• trommarinn = the drummer

So -inn here is the masculine singular definite ending. This is one of the first big differences from English: Icelandic often says noun + the as one word.

The verb brosa often uses the preposition til when you say who someone is smiling at.

So:

• brosa = to smile
• brosa til einhvers = to smile at someone

The preposition til requires the genitive case, which is why söngkonan changes form:

• söngkonan = the singer (nominative)
• söngkonunnar = of the singer / to the singer after til (genitive)

So til söngkonunnar is not a random form; it is there because til governs the genitive.

Because several things are happening at once:

• the base noun is söngkona = female singer
• it is definite = the singer
• it is in the genitive singular
• feminine nouns can change quite a lot across cases

So söngkonunnar is basically the form that fits after til.

This is very normal in Icelandic. A noun can look quite different depending on:

• case
• number
• definiteness
• gender

Á meðan means while or during the time that.

It introduces a clause describing something happening at the same time as the action in the main clause:

• main clause: Trommarinn brosir til söngkonunnar
• time clause: á meðan hún lagar fiðluna sína

So the structure is basically X happens while Y happens.

Most naturally, it refers to the singer.

Why?

• hún is a feminine singular pronoun: she
• söngkonan is a female person
• trommarinn is masculine here, so hún would not normally refer to him

So the most likely reading is that the drummer smiles at the singer while she fixes her violin.

In a larger context, a pronoun can sometimes be ambiguous, but in this sentence the singer is the natural referent.

This is a very important Icelandic point.

Sinn / sín / sitt is a reflexive possessive, used when the possession belongs to the subject of the same clause.

Here the subject of the subordinate clause is hún, so:

• hún lagar fiðluna sína = she fixes her own violin

If you said:

• hún lagar fiðluna hennar

that would usually mean:

• she fixes her violin, where her refers to some other female person, not the subject hún

So Icelandic makes a distinction that English usually does not make clearly.

Because sinn has to agree with the noun it belongs to.

The noun is fiðluna, which is:

• feminine
• singular
• accusative
• definite

So the matching form is sína.

You can think of it like this:

• sinn = masculine forms
• sín / sína = feminine forms depending on case
• sitt = neuter forms

Here the grammar of fiðluna determines the form sína.

Because fiðluna is the direct object of lagar.

The verb laga usually takes an object in the accusative case. So:

• fiðla = violin
• fiðlan = the violin (nominative)
• fiðluna = the violin (accusative)

Since she is fixing the violin, Icelandic uses the accusative object form fiðluna.

Lagar is the present tense of laga.

Depending on context, laga can mean things like:

• fix
• repair
• adjust
• put right

With something like a violin, it could suggest repairing it, adjusting it, or getting it into proper condition. The exact nuance depends on context, but grammatically it is simply the present tense form: is fixing / fixes.

Yes, but it is also very normal Icelandic word order.

The main clause is:

• Trommarinn brosir til söngkonunnar

That follows the common pattern where the finite verb comes early in the clause.

Then the subordinate clause comes after á meðan:

• hún lagar fiðluna sína

If you moved the á meðan clause to the front, Icelandic would still keep its normal verb placement in the main clause:

• Á meðan hún lagar fiðluna sína brosir trommarinn til söngkonunnar.

So word order is flexible, but the verb-position rules still matter.

Yes.

• söngkona = female singer
• a more general or masculine form would be different, such as söngvari

So this sentence explicitly tells you that the singer is female, which is one reason the later pronoun hún fits so naturally.

In natural English translation, smile at is usually the best match here.

Even though til often has a literal sense like to or toward, the expression brosa til einhvers idiomatically means to smile at someone or smile toward someone.

So it is better to learn brosa til as a set pattern rather than translating til word-for-word every time.